Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 42yzjvkvβαixγαβjyxiTheorem 3.9 The direction cosines of a nonzero vector v = v 1 i+v 2 j+v 3 k arecosα = v 1‖v‖ , cosβ = v 2‖v‖ , cosγ = v 3‖v‖The direction cosines of a vector v = v 1 i+v 2 j+v 3 k can be computed by normalizingv and reading off the components of v/‖v‖, sincev‖v‖ = v 1‖v‖ i+ v 2‖v‖ j+ v k‖v‖ k = (cosα)i+(cosβ)j+(cosγ)kMoreover, we can show that the direction cosines of a vector satisfy the equationcos 2 α+cos 2 β +cos 2 γ = 1Example 3.15 Find the direction angles of the vector v = 4i−5j+3k.Solution .........Decomposing Vectors into Orthogonal ComponentsOur next objective is to develop a computational procedure for decomposing a vector intosum of orthogonal vectors. For this purpose, suppose that e 1 and e 2 are two orthogonalunit vectors in 2-space, and suppose that we want to express a given vector v as a sumv = w 1 +w 2so that w 1 is a scalar multiple of e 1 and w 2 is a scalar multiple of e 2 .w 2e 2ve 1 w 1